The design and management of complex systems often involve the reconstruction of patterned phenomena. In order to better design cellular telephone networks, for example, engineers need to have accurate approximations of transmission tower signals at all points within the network's coverage area. Other examples may include computer packet routing software that require accurate estimations of routing times to every node within the network, or atmospheric prediction models that require accurate estimations of temperatures at all points covered by the models, or cell density estimations for tissue samples. In order to design and implement applications in these, and like systems, designers often rely on their knowledge of the nature of the phenomena upon which those applications are based.
Most applications do not involve phenomena that can be perfectly reconstructed, with absolute precision and certainty, at every point within the measurement space, but do involve variables capable of being measured. Many methods have been developed that approximate or estimate the value of a phenomenon from those measurements within a limited region. In traditional methods, designers assume that the phenomenon behaves as a polynomial. Through continued measurements, such a method will continue to refine the approximation, but will retain the assumption that the underlying phenomenon values are polynomic through the measurement space.
Although the most common assumption is that phenomena mimic polynomials, an understanding of what generated the phenomena may lead to a trigonometric model, an exponential model, or some other appropriate mathematical prediction of value. Typically, reconstruction using these models will take the form of a measurement system that attempts to collect enough points to satisfy the function type assumption, and to further isolate any unknown parameters of that function. For example, spectrum analysis assumes that a signal can be decomposed into some combination of sine and cosine functions. Using a spectrum analyzer, the phenomenon being reconstructed is then measured and the appropriate coefficients for the sine and cosine functions are determined through continued measurement. When the data is plotted on top of the prediction, the functional form will then approximate the data. Residuals may also be calculated that give an estimate of how well the measured data fits the predicted functional form.